Geometry of Spaces of Planar Quadrilaterals

Abstract

The purpose of this dissertation is to investigate the geometry of spaces of planar quadrilaterals. The topology of moduli spaces of planar quadrilaterals (the set of all distinct planar quadrilaterals with fixed side lengths) has been well-studied [5], [8], [10]. The symplectic geometry of these spaces has been studied by Kapovich and Millson [6], but the Riemannian geometry of these spaces has not been thoroughly examined. We study paths in the moduli space and the pre-moduli space. We compare intraplanar paths between points in the moduli space to extraplanar paths between those same points. We give conditions on side lengths to guarantee that intraplanar motion is shorter between some points. Direct applications of this result could be applied to motion-planning of a robot arm. We show that horizontal lifts to the pre-moduli space of paths in the moduli space can exhibit holonomy. We determine exactly which collections of side lengths allow holonomy.